Answer :
Certainly! Let's solve the problem step-by-step.
### Part (i): Find the base radius of the cone.
When a sector of a circle is used to form a cone, the arc length of the sector becomes the circumference of the base of the cone.
Given:
- The radius of the sector [tex]\( r_{\text{sector}} \)[/tex] is [tex]\( 7 \)[/tex] cm.
- The angle of the sector [tex]\( \theta \)[/tex] is [tex]\( 270^\circ \)[/tex].
- The value of [tex]\( \pi \)[/tex] is given as [tex]\( \frac{22}{7} \)[/tex].
1. Calculate the length of the arc of the sector:
The formula for the arc length [tex]\( L \)[/tex] of a sector is:
[tex]\[ L = \left(\frac{\theta}{360^\circ}\right) \times 2\pi r_{\text{sector}} \][/tex]
Plugging in the given values:
[tex]\[ L = \left(\frac{270^\circ}{360^\circ}\right) \times 2 \times \frac{22}{7} \times 7 \, \text{cm} \][/tex]
Simplifying this:
[tex]\[ L = \left(\frac{270}{360}\right) \times 2 \times 22 \, \text{cm} \][/tex]
[tex]\[ L = \left(\frac{3}{4}\right) \times 44 \, \text{cm} \][/tex]
[tex]\[ L = 33 \, \text{cm} \][/tex]
2. Relate the arc length to the circumference of the base of the cone:
The arc length [tex]\( L \)[/tex] is the circumference of the base of the cone, which can be represented by:
[tex]\[ \text{Circumference of base of cone} = 2\pi r_{\text{cone}} \][/tex]
3. Solve for the base radius [tex]\( r_{\text{cone}} \)[/tex]:
[tex]\[ 33 \, \text{cm} = 2\pi r_{\text{cone}} \][/tex]
Therefore,
[tex]\[ r_{\text{cone}} = \frac{33 \, \text{cm}}{2 \times \pi} \][/tex]
Substituting the given value of [tex]\( \pi \)[/tex]:
[tex]\[ r_{\text{cone}} = \frac{33 \, \text{cm}}{2 \times \frac{22}{7}} \][/tex]
Simplifying this:
[tex]\[ r_{\text{cone}} = \frac{33 \times 7}{44} \][/tex]
[tex]\[ r_{\text{cone}} = \frac{231}{44} \][/tex]
[tex]\[ r_{\text{cone}} = 5.25 \, \text{cm} \][/tex]
### Part (ii): Calculate the area of the base of the cone correct to the nearest square centimeter.
Given,
[tex]\[ r_{\text{cone}} = 5.25 \, \text{cm} \][/tex]
We need to find the area of the base using the formula for the area of a circle:
[tex]\[ \text{Area} = \pi r_{\text{cone}}^2 \][/tex]
Substituting the values we have:
[tex]\[ \text{Area} = \frac{22}{7} \times (5.25)^2 \][/tex]
Calculating the square of [tex]\( 5.25 \)[/tex]:
[tex]\[ (5.25)^2 = 27.5625 \][/tex]
Now, subsituting back:
[tex]\[ \text{Area} = \frac{22}{7} \times 27.5625 \][/tex]
[tex]\[ \text{Area} = 86.625 \, \text{cm}^2 \][/tex]
Therefore, the answers are:
1. The base radius of the cone is [tex]\( 5.25 \)[/tex] cm.
2. The area of the base of the cone is [tex]\( 87 \)[/tex] cm² (rounded to the nearest square centimeter).
### Part (i): Find the base radius of the cone.
When a sector of a circle is used to form a cone, the arc length of the sector becomes the circumference of the base of the cone.
Given:
- The radius of the sector [tex]\( r_{\text{sector}} \)[/tex] is [tex]\( 7 \)[/tex] cm.
- The angle of the sector [tex]\( \theta \)[/tex] is [tex]\( 270^\circ \)[/tex].
- The value of [tex]\( \pi \)[/tex] is given as [tex]\( \frac{22}{7} \)[/tex].
1. Calculate the length of the arc of the sector:
The formula for the arc length [tex]\( L \)[/tex] of a sector is:
[tex]\[ L = \left(\frac{\theta}{360^\circ}\right) \times 2\pi r_{\text{sector}} \][/tex]
Plugging in the given values:
[tex]\[ L = \left(\frac{270^\circ}{360^\circ}\right) \times 2 \times \frac{22}{7} \times 7 \, \text{cm} \][/tex]
Simplifying this:
[tex]\[ L = \left(\frac{270}{360}\right) \times 2 \times 22 \, \text{cm} \][/tex]
[tex]\[ L = \left(\frac{3}{4}\right) \times 44 \, \text{cm} \][/tex]
[tex]\[ L = 33 \, \text{cm} \][/tex]
2. Relate the arc length to the circumference of the base of the cone:
The arc length [tex]\( L \)[/tex] is the circumference of the base of the cone, which can be represented by:
[tex]\[ \text{Circumference of base of cone} = 2\pi r_{\text{cone}} \][/tex]
3. Solve for the base radius [tex]\( r_{\text{cone}} \)[/tex]:
[tex]\[ 33 \, \text{cm} = 2\pi r_{\text{cone}} \][/tex]
Therefore,
[tex]\[ r_{\text{cone}} = \frac{33 \, \text{cm}}{2 \times \pi} \][/tex]
Substituting the given value of [tex]\( \pi \)[/tex]:
[tex]\[ r_{\text{cone}} = \frac{33 \, \text{cm}}{2 \times \frac{22}{7}} \][/tex]
Simplifying this:
[tex]\[ r_{\text{cone}} = \frac{33 \times 7}{44} \][/tex]
[tex]\[ r_{\text{cone}} = \frac{231}{44} \][/tex]
[tex]\[ r_{\text{cone}} = 5.25 \, \text{cm} \][/tex]
### Part (ii): Calculate the area of the base of the cone correct to the nearest square centimeter.
Given,
[tex]\[ r_{\text{cone}} = 5.25 \, \text{cm} \][/tex]
We need to find the area of the base using the formula for the area of a circle:
[tex]\[ \text{Area} = \pi r_{\text{cone}}^2 \][/tex]
Substituting the values we have:
[tex]\[ \text{Area} = \frac{22}{7} \times (5.25)^2 \][/tex]
Calculating the square of [tex]\( 5.25 \)[/tex]:
[tex]\[ (5.25)^2 = 27.5625 \][/tex]
Now, subsituting back:
[tex]\[ \text{Area} = \frac{22}{7} \times 27.5625 \][/tex]
[tex]\[ \text{Area} = 86.625 \, \text{cm}^2 \][/tex]
Therefore, the answers are:
1. The base radius of the cone is [tex]\( 5.25 \)[/tex] cm.
2. The area of the base of the cone is [tex]\( 87 \)[/tex] cm² (rounded to the nearest square centimeter).
